Задача №1627

\[ \int\frac{dx}{x\sqrt{x^2+x+1}} =\int\frac{dx}{x\cdot|x|\sqrt{1+\frac{1}{x}+\frac{1}{x^2}}} =\sgn{x}\int\frac{dx}{x^2\sqrt{1+\frac{1}{x}+\frac{1}{x^2}}} =\left[t=\frac{1}{x}\right]=\\ =-\sgn{x}\int\frac{dt}{\sqrt{t^2+t+1}} =-\sgn{x}\int\frac{d\left(t+\frac{1}{2}\right)}{\sqrt{\left(t+\frac{1}{2}\right)^2+\frac{3}{4}}}=\\ =-\sgn{x}\cdot\ln\left(t+\frac{1}{2}+\sqrt{t^2+t+1}\right) =-\sgn{x}\cdot\ln\left(\frac{1}{x}+\frac{1}{2}+\sqrt{1+\frac{1}{x}+\frac{1}{x^2}}\right)+C. \]